Our model for a cascade of two diffusion-trapping systems is given by the following equations:

Equation 1 and Equation 2 describe the concentrations of the extracellular ligand (active Trunk) and ligand-receptor complexes (Torso-Trunk) complexes, denoted by L(x,t) and C(x,t), respectively. Note that the receptor-bound ligand does not diffuse; this reflects the fact that the lateral movement of Torso receptors is highly restricted (personal communication). Equation 3 and Equation 4 describe the concentrations for the cytoplasmic and nuclear signaling molecules (phosphorylated MAPK), that are denoted by S(x,t) and N(x,t), respectively. Extracellular ligand is “injected” into the system at x = 0, with a time dependent flux, Q(t). A combination of ligand diffusion, reversible binding to surface receptors and receptor-mediated internalization establishes a spatial profile of cell surface receptor occupancy, which in turn serves as a source for the intracellular signaling species, which shuttle in and out of the nuclei and can be ‘degraded’ (dephosphorylated) in either of these compartments.
Equation 5 and Equation 6 provide the initial and boundary conditions. The system is modeled as one-dimensional (along the AP axis) and semi-infinite in space, reflecting the fact that the observed dpERK pattern is localized at the poles of the embryo. Receptors are assumed to be in excess and the rate constant of ligand binding, kb, is proportional to receptor expression level. Finally, all signal transduction processes, from the receptors to phosphorylated MAPK, have been lumped into a single constant, g, which characterizes the rate at which the dpERK molecules are generated (indirectly) by a single ligand/receptor complex.
The extracellular part of the model is essentially identical to those used to describe the patterning morphogens in other systems, e.g., the patterning of the follicular epithelium by Gurken or patterning of the wing imaginal disk by Dpp. The intracellular module is similar to the previously published descriptions of phosphorylation gradients in spatially distributed models of signal transduction pathways. Using this model we can explore how the pattern of MAPK phosphorylation is controlled by a combination of localized ligand release, extracellular diffusion, binding to cell surface receptors, receptor-mediated ligand internalization, and diffusion, nuclear trapping and dephosphorylation of activated MAPK inside the embryo.